Problem Bank

The numbers $1,1,1,1,1,2,\dots,2,3,\dots,3,4,\dots,4,5,\dots,5$ (five of each) fill a $5 \times 5$ grid, one per cell, so that any two numbers in the same column differ by at most …

Let $a > b > 0$ and $n \in \mathbb{N}^*$. Define

1. Prove $A_2 > B_2$.…

Sequences $\{a_n\}, \{b_n\}$ satisfy $a_1 = b_1 = 1$ and

Let $S_n$ be the sum of the first $n$ terms of $\{a_n + b_n\}$. Find …

Let $f(x) = \mathrm{e}^x \sin(rx)$ with $r \in \mathbb{N}^*$.

1. If $r = 1$, find the intervals on which $f(x)$ is monotonic. 2. Prove that for every positive real number $M$ ther…

For a function $f$ on $[a, b]$, define its oscillation $d(a, b)$ as the difference between its maximum and minimum there.

1. If $f(x) = x^2 - 2x + 2$, find $d(1, 2)$. 2. If $f(x…

Triangle $A_0B_0C_0$ has interior angles $59.999^\circ$, $60^\circ$, $60.001^\circ$. For each positive integer $n$, let $A_n$ be the foot of the perpendicular from $A_{n-1}$ to sid…

Let $f(x)=\dfrac{1-\ln x}{x^2}$.

1. Find the zeros of $f(x)$ and its intervals of monotonicity. 2. Prove that the curve $y=\dfrac{\ln x}{x}$ has a tangent line of slope $6$, and t…

You may use the fact that for $x \in \left(0, \dfrac{\pi}{2}\right)$,

1. Prove that $\dfrac{\sin x}{x} > \dfrac{1}{2}$ for $x \in \left(0, \d…

Let $a \in \mathbb{R}$ and

If $f(x)$ has exactly $6$ zeros in $(…

A function $f(x)$ on $\mathbb{R}$ is odd with period $6$, and for $x \in (0, 3)$,

If $f(x)$ has exactly $5$ zeros in $[-3, 3]$, find the…

Let $a \in \mathbb{R}$ be such that for every real $b$,

Find the range of $a$.

The function $f(x) = \ln x + \dfrac{m}{x} - 3$ has two zeros.

1. Find the range of possible values of $m$. 2. Let $a, b$ be the two zeros of $f(x)$. Prove that